The empty rooms inside a close-packed crystal
Guides 1 and 2 left you with a clean picture: stack close-packed layers ABCABC or ABAB, and identical hard spheres fill 74 percent of all space — a packing factor of 0.74, coordination 12, the tightest any single sphere size can manage. But read that number the other way round. If 74 percent is solid, then 26 percent is empty, and that empty quarter does not vanish. It is not smeared thinly through the crystal like air in a sponge; it is gathered into small, regularly repeating pockets sitting in fixed spots between the spheres. Those pockets are the interstitial sites, and this guide is entirely about them.
Picture the grocer's orange pyramid from guide 1 one more time. Look down into the top layer and you can see two sorts of dimple: a shallow one where a single orange rests in the crook of three below it, and a deeper channel where the gaps of one layer line up with the gaps of the next. A marble tucked into the first kind touches four oranges; one dropped into the second touches six. Those two dimples, repeated through the whole stack, are the only two hole shapes a close-packed crystal owns — and they turn out to be the real estate on which nearly every compound is built.
That last point is why the holes deserve a whole guide rather than a footnote. In an ionic solid, the big ions almost always do the close packing while the small ions simply move into the holes between them. So rock-salt, fluorite, spinel — the whole zoo of guides 4 and 5 — are just answers to the questions: which holes get filled, and how many of them? Before we can name any of those structures we have to know the holes cold: their shapes, their count, and their exact size.
Two shapes of hole, and how to count them
The two dimples have proper names. The shallow one — a sphere resting in the crook of three, capped by a fourth from the layer above — is a tetrahedral hole: four spheres surround it with their centres at the corners of a tetrahedron, so a guest sitting there has coordination number 4. The deeper one — where a triangle of three spheres pointing up meets a triangle of three pointing down — is an octahedral hole: six spheres, their centres at the corners of an octahedron, coordination 6. Every gap in a close-packed metal is one of these two, nothing else.
Now count them, because the count decides everything downstream. For every N spheres in a close-packed array there are exactly N octahedral holes and 2N tetrahedral holes — twice as many tetrahedral ones, which fits the intuition that the smaller, shallower dimple is easier to come by. Anchor it in the face-centred cubic cell you already know: it holds 4 atoms, so it holds 4 octahedral holes and 8 tetrahedral holes. The octahedral holes sit at the cube's body centre and the midpoint of each of its 12 edges (12 edges shared four ways, 12/4 = 3, plus the one at the centre = 4). The tetrahedral holes sit at the eight (1/4, 1/4, 1/4)-type points, one tucked into each corner octant of the cube.
This little arithmetic is secretly a preview of every ionic structure to come. Take an FCC array of anions and fill *all 4* octahedral holes with cations: you get a 4-to-4 ratio, formula MX — that is rock-salt. Fill half the 8 tetrahedral holes instead: still 4-to-4, MX again, but now the cation is 4-coordinate — that is zinc-blende. Fill *all 8* tetrahedral holes: 8-to-4, formula MX2 — that is fluorite. Same close-packed skeleton, three different occupancy rules, three famous structures. Hold that thought; it is the spine of guide 4.
How big is a hole? Where 0.225 and 0.414 come from
A hole is not just a place; it has a definite size — the radius of the largest sphere you could drop in until it just touches the walls. For spheres of radius R, that turns out to be r = 0.225 R for a tetrahedral hole and r = 0.414 R for an octahedral one. The octahedral hole is roomier, which makes sense: six spheres enclose more empty space than four. These two numbers are small on purpose — a close-packed array is tight, and its leftover rooms are cramped.
The octahedral number is worth deriving once, because it is a single clean line. In FCC the spheres touch along a face diagonal, so 4R = a times sqrt(2), giving R = a / (2 times sqrt(2)). The octahedral hole at the cube's centre lies a distance a/2 from each of the six face-centre atoms, so R + r = a/2. Subtract: r = a/2 - a/(2 times sqrt(2)) = (a/2)(1 - 1/sqrt(2)). Divide by R and the cell edge cancels: r/R = sqrt(2) - 1 = 0.414. The tetrahedral case runs the same way and lands on r/R = sqrt(6)/2 - 1 = 0.225.
The radius-ratio rule
Now flip the question around. Instead of asking how big a hole is, ask: given a small cation of radius r+ and a large anion of radius r-, which hole does the cation want to live in? Electrostatics gives a clear pull in two directions. A cation lowers its energy by touching as many anions as possible, so it wants a high coordination number. But those anions must genuinely touch it — a cation rattling loose in an oversized hole has anion neighbours drifting away, which costs energy. So the cation settles into the largest hole it can fill snugly without rattling. Comparing r+/r- against the hole sizes we just derived tells you which hole that is. This is the radius-ratio rule.
- Identify the smaller ion (usually the cation) and the larger (usually the anion), and look up their ionic radii — but use radii quoted for the coordination you expect, because an ion's tabulated radius grows as its coordination rises.
- Form the ratio r+/r-, a single number between 0 and 1.
- Read off the band: 0.225 to 0.414 means a tetrahedral hole (coordination 4); 0.414 to 0.732 means octahedral (coordination 6); 0.732 to 1.000 means a cubic hole (coordination 8).
- That coordination predicts which hole the cation occupies and hints at the whole structure type. Then check it against the real crystal — and treat any ratio sitting near a boundary as a coin toss, not a verdict.
RADIUS-RATIO RULE r+ / r- -> hole shape, coordination number
r+/r- range cation sits in neighbours example
--------------- ----------------- ---------- -------------
0.155 - 0.225 triangular hole 3 (uncommon)
0.225 - 0.414 TETRAHEDRAL hole 4 Zn in ZnS
0.414 - 0.732 OCTAHEDRAL hole 6 Na in NaCl
0.732 - 1.000 cubic hole 8 Cs in CsCl
1.000 same-size site 12 metals (CCP/HCP)
the two key boundaries ARE the hole sizes in a close-packed array:
tetrahedral hole r/R = sqrt(6)/2 - 1 = 0.225
octahedral hole r/R = sqrt(2) - 1 = 0.414
Work one real case. In common salt the ions are Na+ with r+ = 1.02 angstrom (its 6-coordinate radius) and Cl- with r- = 1.81 angstrom. The ratio is 1.02 / 1.81 = 0.56, which lands squarely in the 0.414-to-0.732 band, so the rule predicts an octahedral hole with coordination 6. And that is exactly right: in the rock-salt structure each Na+ sits in an octahedral hole of an FCC Cl- array, touching six chlorides — the very structure guide 4 opens with. One ratio, one lookup, and you have predicted a crystal.
Pauling's rules: the radius ratio grows up
The radius-ratio idea is really just the first of five empirical guidelines Linus Pauling distilled in the 1920s for making sense of ionic crystals — Pauling's rules. Rule 1, the coordination principle, is exactly what we have been doing: around each cation forms a coordination polyhedron of anions; the cation-anion distance is the sum of their radii; and the number of corners on that polyhedron — the coordination number — is set by the radius ratio. Everything in the previous section is Pauling's first rule in disguise.
Rule 2, the electrostatic valence principle, is the powerful partner and worth carrying with you. Define a bond's strength as s = z / CN, the cation's charge divided by its coordination number. The rule says that the strengths of all the bonds reaching a given anion should add up to that anion's own charge — local electrical neutrality, checked ion by ion. Test it on NaCl: sodium is +1 with coordination 6, so each Na-Cl bond has strength 1/6; each Cl- is touched by six sodiums, and 6 times 1/6 = 1, exactly the magnitude of the chloride charge. It balances. This one arithmetic check is a genuinely useful sanity test for any structure you propose.
The remaining three rules are quicker. Rule 3: polyhedra prefer to share corners rather than edges, and edges rather than faces, because sharing an edge or face drags the two central cations closer and their like charges repel — so shared edges and faces lower stability. Rule 4: cations of high charge and low coordination especially avoid sharing polyhedron elements. Rule 5, the rule of parsimony: a crystal tends to use only a small number of genuinely different kinds of site. Be honest, though — these are empirical guidelines with real exceptions; modern surveys of thousands of oxides find rule 2 holds up well but rules 3 to 5 are broken fairly often. They are a way to reason about a structure, not laws it must obey.
That is the whole toolkit for the two guides ahead. With holes, coordination number, and the radius ratio in hand, guide 4 walks the canonical ionic structures — NaCl filling all the octahedral holes, fluorite filling all the tetrahedral ones, cesium chloride abandoning close packing altogether for cubic 8-coordination, then perovskite and spinel. Guide 5 turns to the covalent solids — diamond and zinc-blende, where cations again take half the tetrahedral holes, but where directional covalent bonds, not sphere-packing, ultimately call the shots. In every one, the first move is the same: find the holes, and see who moved in.